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LEAP-Asia-2018 Numerical Simulation Exercise – Phase I
Model Calibration report
Gianluca Fasano, Anna Chiaradonna, Emilio Bilotta
University of Napoli Federico II
January, 2019
ii
Table of Contents
Introduction ....................................................................................................................................... 3
Model Description ......................................................................................................................... 3
Model Parameters .......................................................................................................................... 6
Calibration Method ....................................................................................................................... 7
Liquefaction Strength Curves ........................................................................................................ 8
Simulation Results ............................................................................................................................ 9
References ....................................................................................................................................... 10
Appendix A: Simulation cyclic torsional tests Dr = 50% ............................................................... 11
Appendix B: Simulation cyclic torsional tests Dr = 60% ............................................................... 15
Introduction This report describes the process followed in the calibration of the selected constitutive
model. The model calibration report covers the essential features of the constitutive model, the
final model parameters, the calibration philosophy and the assumptions used in the calibration
process. Finally, the report also presents a comparison between the predicted and experimental
cyclic laboratory tests and liquefaction resistance curves.
The adopted constitutive model has been calibrated on the results of the provided cyclic
torsional shear tests for Dr = 50% and 60% under an initial effective confining stress of 100 kPa.
Model Description The constitutive model used in the simulation exercise is the PM4Sand model (Boulanger
and Ziotopoulou 2015). The PM4Sand (version 3.1) model follows the basic framework of the
stress-ratio controlled, critical state compatible, bounding surface plasticity model for sands
presented by Dafalias and Manzari (2004), who extended the previous work by Manzari and
Dafalias (1997) by adding a fabric-dilatancy related tensor quantity to account for the effect of
fabric changes during loading. The fabric-dilatancy related tensor was used to macroscopically
model the effect that microscopically-observed changes in sand fabric during plastic dilation have
on the contractive response upon reversal of loading direction. The modifications were developed
and implemented to improve the ability of the model to match existing engineering design
relationships currently used to estimate liquefaction-induced ground deformations during
earthquakes. These modifications are described in the manuals (version 1 in Boulanger 2010,
version 2 in Boulanger and Ziotopoulou 2012, and version 3) and in the associated publications,
as listed in the mentioned manuals.
The model is written in terms of effective stresses, with the conventional prime symbol
dropped from the stress terms for convenience because all stresses are effective for the model.
The stresses are represented by the tensor r, the principal effective stresses are 1, 2, and 3,
the mean effective stress is p, the deviatoric stress tensor is s, and the deviatoric stress ratio tensor,
r. The current implementation was further simplified by casting the various equations and
relationships in terms of the in-plane stresses only. This limits the implementation to plane-strain
(2D) applications, having the further advantage in its simplified implementation to improve the
computational speed. The relationships between the various stress terms can be summarized as
follows:
= (𝑥𝑥 𝑥𝑦
𝑥𝑦 𝑦𝑦) (1)
𝑝 =𝑥𝑥 + 𝑦𝑦
2 (2)
𝒔 = − 𝑝𝑰 = (𝑠𝑥𝑥 𝑠𝑥𝑦
𝑠𝑥𝑦 𝑠𝑦𝑦) = (
𝑥𝑥 − 𝑝 𝑥𝑦
𝑥𝑦 𝑦𝑦 − 𝑝) (3)
𝒓 =𝒔
𝑝= (
𝑟𝑥𝑥 𝑟𝑥𝑦
𝑟𝑥𝑦 𝑟𝑦𝑦) = (
𝑥𝑥 − 𝑝
𝑝
𝑥𝑦
𝑝𝑥𝑦
𝑝
𝑦𝑦 − 𝑝
𝑝
) (4)
In eq. (3), 𝑰 is the identity matrix. The deviatoric stress and deviatoric stress ratio tensors
are symmetric with 𝑟𝑥𝑥 = −𝑟𝑦𝑦 and 𝑠𝑥𝑥 = −𝑠𝑦𝑦 (meaning a zero trace).
The strains are represented by a tensor, , expressed as the sum of the volumetric strain 𝑣
and of the deviatoric strain tensor, 𝑒. The volumetric strain is,
𝜀𝑣 = 𝜀𝑥𝑥 + 𝜀𝑦𝑦 (5)
and the deviatoric strain tensor is,
𝒆 = −𝑣
3𝑰 = (
𝑥𝑥 −𝑣
3𝑥𝑦
𝑥𝑦 𝑦𝑦 −𝑣
3
) (6)
In incremental form, the deviatoric and volumetric strain terms are decomposed into an
elastic and a plastic part,
𝑑𝒆 = 𝑑𝒆𝑒𝑙 + 𝑑𝒆𝑝𝑙 (7)
𝑑𝑣 = 𝑑𝑣𝑒𝑙 + 𝑑𝑣
𝑝𝑙 (8)
where:
𝑑𝒆 𝑒𝑙 = elastic deviatoric strain increment tensor
𝑑𝒆 𝑝𝑙 = plastic deviatoric strain increment tensor
𝑑 𝑣 𝑒𝑙 = elastic volumetric strain increment tensor
𝑑 𝑣 𝑝𝑙
= plastic volumetric strain increment tensor
This constitutive model follows the critical state theory and uses the relative state parameter
index (𝑅
) as defined by Boulanger (2010) and shown in Figure 1. This relative parameter is
defined by an empirical relationship for the critical state line:
𝑅
= 𝐷𝑅,𝑐𝑠 − 𝐷𝑅 (9)
𝐷𝑅,𝑐𝑠 =𝑅
𝑄 − ln (100𝑝
𝑝𝐴)
(10)
where 𝐷𝑅,𝑐𝑠 is the relative density at critical state for the current mean effective stress,
instead, 𝑄 and 𝑅 are two parameters that define the shape of critical curve.
Figure 1: Relative state parameter index
Bounding, dilatancy and critical surfaces are incorporated in PM4Sand following the form
of Dafalias and Manzari (2004).
The bounding (𝑀𝑏) and dilatancy (𝑀𝑑) ratios can be related to the critical stress (𝑀) ratio:
𝑀𝑏 = 𝑀 exp(−𝑛𝑏𝑅
) (11)
𝑀𝑑 = 𝑀 exp(−𝑛𝑑𝑅
) (12)
where 𝑛𝑏 and 𝑛𝑑 are model parameters. The relationship for 𝑀 is:
𝑀 = 2 sin(𝑐𝑣
) (13)
where 𝑐𝑣
is critical state friction angle.
As the soil is sheared toward critical state (𝑅
= 0), the values of 𝑀𝑏 and 𝑀𝑑 will both
approach the value of M. Thus, the bounding and dilatancy surfaces move together during
shearing until they coincide with the critical state surface when the soil has reached critical state.
The few experimental data for loose-of-critical sands (having no peak) show that the
maximum friction angles (presumably determined at the limit of strains possible within the
laboratory tests) were only slightly smaller than the critical state values, such that extending the
above relationships to loose-of-critical sands may tend to underestimate the peak friction angles
(in this case theoretically coinciding with the critical state one). Consequently, in the present
formulation the model allows 𝑛𝑏 and 𝑛𝑑 to be different for loose-of-critical and dense-of-critical
states for the same sand.
A large portion of the post-liquefaction reconsolidation strains are due to the sedimentation
effects which are not easily incorporated into either the elastic or plastic components of
behaviour. For this reason, in the PM4Sand a post-shaking function was implemented. In a
strongly pragmatic way, this function reduces volumetric and shear moduli, thus increasing
reconsolidation strains to somehow simulate the sedimentation ones (not included in the model).
The post-shaking elastic moduli are determined by multiplying the conventional elastic
moduli by a reduction factor 𝐹𝑠𝑒𝑑 as,
𝐺𝑝𝑜𝑠𝑡−𝑠ℎ𝑎𝑘𝑖𝑛𝑔 = 𝐹𝑠𝑒𝑑𝐺 (14)
𝐾𝑝𝑜𝑠𝑡−𝑠ℎ𝑎𝑘𝑖𝑛𝑔 = 𝐹𝑠𝑒𝑑𝐾 (15)
for more information on the 𝐹𝑠𝑒𝑑 it is possible refer to Boulanger and Ziotopoulou (2015).
The model require 27 input parameters, 3 of these are considered primary parameters while
all the other parameters are suggested to be left with their default values. Table 1 reports the most
important input parameters of the PM4Sand model, which were defined in the calibration process.
Table 1: Input parameters of the PM4Sand model
Dr Initial relative density
G0 shear modulus coefficient
hp0 contraction rate parameter
pA atmospheric pressure
emax maximum void ratio
emin minimum void ratio
nb bounding surface parameter
nd dilatancy surface parameter
φcv critical state friction angle
Poisson's ratio
Q critical state line parameter
R critical state line parameter
Model Parameters
The model parameters obtained from the calibration process are listed in Table 2, which also
include some parameters kept at their default value.
The model parameters are obtained by using the results of the provided cyclic torsional shear
tests, as described in the next paragraph about the calibration procedure. It should also be noted
that the selected model parameters from this report will be adopted for future simulations of the
centrifuge model tests.
Table 2: Parameters of the PM4Sand model based on the cyclic torsional test data
Initial relative
density
Model
parameters
Dr = 50% Dr = 60%
Dr 0.5 0.6
G0 630 730
hp0 0.08 0.05
pA 101.3 101.3
emax 0.78 0.78
emin 0.51 0.51
nb 0.5 0.5
nd 0.3 0.1
φcv 32 32
0.3 0.3
Q 10 10
R 1 1
PostShake 0 0
Calibration Method
The approach used in the calibration of the constitutive model parameters is hereafter
explained.
The PM4Sand constitutive model is calibrated on the results of laboratory element tests.
PM4Sand has 27 input parameters (6 primary and 21 secondary) but only three of them are
required as independent inputs: the initial relative density (Dr), the shear modulus coefficient used
to define the small strain shear modulus (Go) and the contraction rate parameter used for the
calibration of the undrained shear strength (hp0). Basically, these three parameters were calibrated
against the experimental data. The initial relative density has been set equal the value of relative
density used in the cyclic torsional tests, Dr=0.5 and 0.6.
The value of the shear modulus coefficient Go was determined as a function of the relative
density using the follow relationship:
𝐺𝑜 = 167 ∙ √46 ∙ 𝐷𝑟2 + 2.5 (16)
The parameter hp0 scales the plastic contraction rate and is the primary parameter for the
calibration of undrained cyclic strength. It is calibrated using an iterative process, in which
undrained single-element simulations are conducted to match with the experimental liquefaction
triggering curve by keeping the other parameters fixed.
With reference to the secondary parameters of the model, some with a clear physical meaning
have been defined on the available experimental data, while the others have been left with their
default values.
Shear strength parameters are computed from the monotonic triaxial test data, available on the
NEES Hub (https://nees.org/dataviewer/view/1064:ds/1189).
Drained triaxial compression tests, carried out by Vasko (2015) on loose and dense specimens,
were used to define the critical state line in the plane q : p’ and the constant volume friction angle,
’c. As well known, the evaluation of critical state conditions in triaxial tests is a very complex
issue, being such a test intrinsically affected by a number of experimental limitations (localization,
bulging, shear stresses on the rough porous stones, difference between local and external
displacements, etc.). One of the best ways to evaluate the final state is therefore the one that
analyses dilatancy trend at the end of the tests. Based on all the elaborations of the available
experimental data, and considering that the hypoplastic model assumes the critical state as an
asymptotic state at infinite strains, in this case the best fit of this parameter is the following:
c 32 (17)
Minimum and maximum void ratios, emax and emin, have been defined as mean values of
the experimental measurements carried out in the LEAP-UCD-2017 Simulation Exercise
(Manzari et al. 2019).
To sum up, the model parameters for static loading conditions were defined on the physical
properties and tests results provided for the considered sand.
Conversely, the model parameters for cyclic loading conditions were defined using
experimental data of cyclic torsional tests (Dr = 50% and 60%).
The material parameters used to perform the simulations are those reported in Table 2 for each
relative density. Every cyclic test is simulated imposing the prescribed CSR and computing the
number of cycles, NL, to induce liquefaction. Liquefaction condition has been defined according
to the stress-based approach, i.e., ru=95%, where ru is the excess pore pressure ratio (ru = u/ p’0
ratio between the excess pore water pressure increment induced by cyclic loading and the initial
effective confining pressure applied during the test, p’0).
Liquefaction Strength Curves
The liquefaction strength curves, obtained from the simulated cyclic torsional tests, are
hereafter plotted and compared with the experimental results (Figure 2). Table 3 reports the
numerical values of the simulation results, i.e. the cyclic stress ratio, CSR, versus the number of
cycles until excess pore pressure ratio, ru = u/p’0, achieved 95% for each simulated test.
It can be observed how the adopted calibration provides a good prediction of the experimental
cyclic resistance curve for high/medium values of the cyclic resistance ratio (CRR), while
underestimation of the experimental cyclic strength is observed for low values of CRR.
Figure 2: Liquefaction Strength Curves obtained from experimental and simulated cyclic
torsional tests on Ottawa F65 Sand
Table 3: Predicted liquefaction strength curves from cyclic torsional test
Dr (%) CSR No. of cycles to 95% ru
50 0.099 74
50 0.127 25.5
50 0.149 11.6
50 0.191 5
60 0.117 65.5
60 0.125 51.5
60 0.144 27.5
60 0.174 13
60 0.199 7.5
Simulation Results
In addition to the report on model calibration, the results of the simulation of the cyclic
torsional tests previously mentioned are reported in separate files with the required format, i.e.
excel files. The results of element test simulations and comparison with those of the provided
cyclic torsional tests are also reported in Appendix A and B for Dr = 50 % and 60%, respectively.
References
Boulanger, R.W. (2010). A sand plasticity model for earthquake engineering applications, Report
No. UCD/CGM-10-01. Technical report, Center for Geotechnical Modeling, Department
of Civil and Environmental Engineering, University of California.
Boulanger, R.W. and Ziotopoulou K. (2012). PM4Sand (Version 2): A sand plasticity model for
earthquake engineering applications, Report No. UCD/CGM-12/01. Technical report,
Center for Geotechnical Modeling, Department of Civil and Environmental Engineering
College of Engineering, University of California at Davis.
Boulanger, R.W. and Ziotopoulou K. (2015). PM4Sand (Version 3): A sand plasticity model for
earthquake engineering applications, Report No. UCD/CGM-15/01. Technical report,
Center for Geotechnical Modeling, Department of Civil and Environmental Engineering
College of Engineering, University of California at Davis.
Dafalias, Y. F., and Manzari, M. T. (2004). Simple Plasticity Sand Model Accounting for Fabric
Change Effects.” ASCE, Journal of Geotechnical Engineering Division, 130(6): 622–634.
Manzari M.T., El Ghoraiby M., Zeghal M., Kutter B.L., Arduino P., Barrero A.R., Bilotta E.,
Chen L., Chen R., Chiaradonna A., Elgamal A., Fasano G., Fukutake K., Fuentes W.,
Ghofrani A., Ichii K., Kiriyama T., Lascarro C., Mercado V., Montgomery J., Ozutsumi
O., Qiu Z., Taiebat M., Travasarou T., Tsiaousi D., Ueda K., Ugalde J., Wada T., Wang
R., Yang M., Zhang J., Ziotopoulou K. (2019). LEAP-2017 Simulation Exercise:
Calibration of Constitutive Models and Simulation of the Element Tests. Proc. of the
LEAP-UCD-2017 workshop. Springer (under pubblication).
Manzari, M. T., and Dafalias, Y. F. (1997). A critical state two-surface plasticity model for sands.
Géotechnique, 47(2): 255–272.
Appendix A: Simulation cyclic torsional tests Dr = 50%
Test 1: CSR = 0.099; Number of cycles until ru = 95% is achieved =74
Test 2: CSR = 0.127; Number of cycles until ru = 95% is achieved =25.5
Test 3: CSR = 0.149; Number of cycles until ru = 95% is achieved =11.6
Test 4: CSR = 0.191; Number of cycles until ru = 95% is achieved =5
Appendix B: Simulation cyclic torsional tests Dr = 60%
Test 1: CSR = 0.117; Number of cycles until ru = 95% is achieved =65.5
Test 2: CSR = 0.125; Number of cycles until ru = 95% is achieved =51.5
Test 3: CSR = 0.144; Number of cycles until ru = 95% is achieved =27.5
Test 4: CSR = 0.174; Number of cycles until ru = 95% is achieved =13
Test 5: CSR = 0.199; Number of cycles until ru = 95% is achieved =7.5
